Purpose of Algebra is to Solve Equations

by Ron Kurtus (revised 10 October 2014)

There are numerous mathematical relationships that have been established in science, finance and other areas. Examples include the relationship between force and acceleration, conversion of centimeters to inches, and determining the payments on a loan with a given interest rate.

The purpose of Algebra is to make it easy to state the relationship by using letters to represent entities as a form of shorthand. The relationship is then called an equation. Algebra then allows you to substitute values for entities in order to determine the value of other items. Finally, you can manipulate the equations to solve for the various items or unknown quantities.

Questions you may have include:

How are letters used to state a relationship?

How are values substituted?

How are equations manipulated?

This lesson will answer those questions.

Stating relationship with letters

The best way to explain how a relationship can be stated with letters is to provide some examples.

Changing inches to centimeters

The relationship for changing inches to centimeters to is to multiple the number of inches by 2.54 to get the number of centimeters.

Suppose we represent inches with the letter i and centimeters with the letter c. Then we have the equation:

c = 2.54 × i

The letters i and c are called variables in the equation, since they can vary depending on what value you give them.

Note: Often, you will see the multiplication sign as * instead of ×, as a way to avoid confusion with x. This is mostly seen in web pages and on some calculators.

Example of loan payments

The amount of interest you pay on a loan is equal to the interest rate times the money owed.

Let:

I = amount of interest (note that this is different than the small i in the previous example)

r = interest rate

m = money owed

Thus, the equation is:

I = r × m

The letters I, r and m are called variables in the equation, since they can vary depending on what value you give them.

Substituting values

Given a relationship, you can substitute values to solve the equation.

Since we have the equation c = 2.54 × i, you can substitute i = 2 inches to find the corresponding number of centimeters.

c= 2.54 × 2

c = 5.08 centimeters

Likewise, since you have the equation I = r × m, and if r = 5% = 0.05 and m = $1000, you can substitute those values into the equation to get:

I = 0.05 × $1000 = $50

Manipulating equations

The equation for converting centimeters into inches can be manipulated to get another equation that converts inches into centimeters by simply dividing both sides of the equal sign by 2.54.

c = 2.54 × i

i = c ÷ 2.54 = c/2.54

Likewise, the equation for interest can be manipulated to be the equation for rate or the equation for money owed:

I = r × m

r = I/m

m = I/r

Summary

The primary purpose of Algebra is to allow you to substitute letters for the names of items in a relationship, thus creating an equation. Then you can substitute in values to solve for an item. You can manipulate the equation to put it in terms of one of the variables.